The broad goals of this project are to develop and extend the dynamic analysis and application of mathematical models of human populations. The investigations proposed here involve studies of three overlapping classes of demographic models. The first class of models incorporates economic feedback between the past composition of a population and future fertility into an intrinsically controlled dynamic model for population. It is porposed to study the non-linear properties of sustained oscillations in such models, especially their form, period, and stability as a function of the nature of the underlying control mechanism. The parametric sensitivity and robustness of these cycles in such variants of the model as the cohort, period, and labor force participation models will be studied and the results applied to parameter estimation and data analysis. The limits to consistency between discrete time and continuous time non linear models will be explored. The impact of external stochastic variation in vital rates, and of externally driven nonstationarity in model parameters, will be investigated. In contrast with this approach two novel methods, threshold autoregressive models and construction of chaotic attractors, will be used to generate non-linear models for population directly from data; the latter models will be compared with economically based control models. A second set of investigations concerns stochastic variation in vital rates: populations near stationarity will be studied using a second order autoregressive fertility process and applying results concerning matrix products; the effect of random environmental variability on long run population behavior will be studied in connection with arguments concerning homeostasis in human history; and the simultaneous effects of external stochastic variation and internal stochasticity will be explored for age structured discrete time models. The final proposed work will investigate two-sex models with fixed vital rates, focusing on the dynamic consequences of the structure of marriage formation rules.